The Riesz transform and quantitative rectifiability for general Radon measures

Abstract

In this paper we show that if $μ$ is a Borel measure in $\mathbb R^{n+1}$ with growth of order $n$, so that the $n$-dimensional Riesz transform $R_μ$ is bounded in $L^2(μ)$, and $B\subset\mathbb R^{n+1}$ is a ball with $μ(B)\approx r(B)^n$ such that: (a) there is some $n$-plane $L$ passing through the center of $B$ such that for some $δ>0$ small enough, it holds $\int_B \frac{dist(x,L)}{r(B)}\,dμ(x)\leq δ\,μ(B),$ (b) for some constant $ε>0$ small enough, $\int_B |R_\mu1(x) - m_{μ,B}(R_\mu1)|^2\,dμ(x) \leq ε\,μ(B)$, where $m_{μ,B}(R_\mu1)$ stands for the mean of $R_\mu1$ on $B$ with respect to $μ$; then there exists a uniformly $n$-rectifiable subset $Γ$, with $μ(Γ\cap B)\gtrsim μ(B)$, and so that $μ|_Γ$ is absolutely continuous with respect to $H^n|_Γ$. This result is an essential tool to solve an old question on a two phase problem for harmonic measure in a subsequent paper by Azzam, Mourgoglou and Tolsa.